Geometric representation theory and moduli spaces of bundles

这个建议属于几何表示理论领域。拟进行的研究主要包括4个课题；它们的统一主题是表示理论与代数曲线和代数曲面上束模空间几何的关系。***第一部分是关于局部域上仿射Kac-Moody群的Hecke代数（这个课题是由A.Braverman和D.Kazhdan提出的，我们建议进一步研究一些问题）。特别是，我们计划开发双仿射Hecke代数的Kazhdan-Lusztig多项式理论的模拟。***在第二部分中，我们提出了局部l函数的一个新的统一几何定义，用于约化群G在具有正特征的局部域上的表示（附属于Langlands对偶群的任意有限维表示）。Godement和Jacquet的定义遵循经典的模式,但相关的定义Schwartz空间更相关,它是基于几何的捆的无限维度空间附加到循环群g计划表明,我们的建设是明确的和它是兼容一些已知结果在本地Langlands通信(如Lusztig classificiation单能性的表示)。***在第三部分中，我们提出了三维N=4超对称规范理论的库仑分支的数学定义及其量子化。该定义基于相应规范群的仿射格拉斯曼的几何形状。我们计划将我们的构造应用于Braden， Licata， Proudfoot和Webster提出的所谓“辛对偶”的推测式直言版本的证明。***在第四部分中，我们提出了简单有限维仿射李代数的Kazhdan-Lusztig猜想的一个新的证明（特别地，涵盖了一些新的情况-例如仿射李代数在临界水平上的表示的情况）。提出的证明是基于所谓的Zastava空间（在有限维情况下）和Uhlenbeck束空间（在仿射情况下）的几何。********

The proposal lies in the field of geometric representation theory. The proposed research essentially consists of 4 subjects; the unifying theme for all of them is the relation between representation theory and geometry of moduli space of bundles on algebraic curves and algebraic surfaces. ***The first part has to do with Hecke algebras for affine Kac-Moody groups over local fields (this subject has been initiated by A.Braverman and D.Kazhdan; we propose to study some further questions. In particular, we plan to develop an analog of the theory of Kazhdan-Lusztig polynomials for double affine Hecke algebras).***In the second part we propose a new uniform geometric definition of local L-functions for representations of a reductive group G over local fields of positive characteristic (attached to an arbitrary finite-dimensional representation of the Langlands dual group) . The definition follows the classical pattern of Godement and Jacquet, but the definition of the relevant Schwartz space is much more involved and it is based on the geometry of perverse sheaves of certain infinite-dimensional spaces attached to the loop group of G. The plan is to show that our construction is well-defined and that it is compatible with some known results on the local Langlands correspondence (such as Lusztig's classificiation of unipotent representations).***In the 3rd part we propose a mathematical definition of the so called Coulomb branch of 3-dimensional N=4 super-symmetric gauge theories and its quantization. The definition is based on the geometry of the affine Grassmannian of the corresponding gauge group. We plan to apply our construction to the proof of the conjectural categorical version of the so called "symplectic duality" due to Braden, Licata, Proudfoot and Webster.***In the 4th part we propose to give a new proof of the Kazhdan-Lusztig conjecture for simple finite-dimensional and affine Lie algebras (in partucular, covering some new cases - such as the case of representations of affine Lie algebras on critical level). The proposed proof is based on the geometry of the so called Zastava spaces (in the finite-dimensional case) and Uhlenbeck spaces of bundles (in the affine case).********